Manuel Krannich: Mapping class groups of highly connected manifolds

Abstract: The classical mapping class group Γ(g) of a surface #ᵍ(S¹ x S¹) of genus g shares many features with its higher dimensional analogue Γ(g,n)—the group of isotopy classes of diffeomorphisms of #ᵍ(Sⁿ x Sⁿ). Some aspects, however, become easier to analyse in high dimensions, which enabled Kreck in the 70’s to describe Γ(g,n) for n>2 in terms of an arithmetic group and the group of exotic spheres. His answer, however, left open two extension problems which were later understood in some dimensions by Crowley, Galatius-Randal-Williams, and Krylov, but remained unsettled in most cases. Motivated by renewed interest in these groups in relation to moduli spaces of manifolds, I will recall Kreck’s description of Γ(g,n) and explain how to resolve the remaining extension problems completely for n odd.